LMFDB - Embedded newform 4368.2.h.q.337.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4368,2,Mod(337,4368)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4368, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0, 0, 1]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("4368.337");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4368 = 2^{4} \cdot 3 \cdot 7 \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4368.h (of order $2$, degree $1$, not minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$34.8786556029$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} + 15x^{6} + 67x^{4} + 77x^{2} + 4 $ x^8 + 15x^6 + 67x^4 + 77*x^2 + 4
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 273)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			337.3
Root			$0.233455i$ of defining polynomial
Character	$\chi$	$=$	4368.337
Dual form			4368.2.h.q.337.6

$q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{3} -2.94550i q^{5} -1.00000i q^{7} +1.00000 q^{9} +O(q^{10})$	$q+1.00000 q^{3} -2.94550i q^{5} -1.00000i q^{7} +1.00000 q^{9} +3.62146i q^{11} +(-3.38801 + 1.23346i) q^{13} -2.94550i q^{15} +1.53309 q^{17} +4.10005i q^{19} -1.00000i q^{21} -7.03387 q^{23} -3.67596 q^{25} +1.00000 q^{27} -3.79095 q^{29} +1.79095i q^{31} +3.62146i q^{33} -2.94550 q^{35} +7.24292i q^{37} +(-3.38801 + 1.23346i) q^{39} +9.15455i q^{41} -5.79095 q^{43} -2.94550i q^{45} -7.25460i q^{47} -1.00000 q^{49} +1.53309 q^{51} -13.3430 q^{53} +10.6670 q^{55} +4.10005i q^{57} +4.84545i q^{59} +2.77601 q^{61} -1.00000i q^{63} +(3.63314 + 9.97937i) q^{65} -14.8248i q^{67} -7.03387 q^{69} -3.62146i q^{71} +14.5670i q^{73} -3.67596 q^{75} +3.62146 q^{77} -6.56696 q^{79} +1.00000 q^{81} +7.25460i q^{83} -4.51571i q^{85} -3.79095 q^{87} -0.636396i q^{89} +(1.23346 + 3.38801i) q^{91} +1.79095i q^{93} +12.0767 q^{95} -11.3240i q^{97} +3.62146i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 8 q^{3} + 8 q^{9}+O(q^{10}) $ 8 * q + 8 * q^3 + 8 * q^9	$ 8 q + 8 q^{3} + 8 q^{9} - 6 q^{13} + 20 q^{17} + 6 q^{23} - 34 q^{25} + 8 q^{27} - 18 q^{29} + 6 q^{35} - 6 q^{39} - 34 q^{43} - 8 q^{49} + 20 q^{51} - 10 q^{53} - 16 q^{55} - 20 q^{61} - 10 q^{65} + 6 q^{69} - 34 q^{75} + 4 q^{77} + 2 q^{79} + 8 q^{81} - 18 q^{87} + 6 q^{91} + 78 q^{95}+O(q^{100}) $ 8 * q + 8 * q^3 + 8 * q^9 - 6 * q^13 + 20 * q^17 + 6 * q^23 - 34 * q^25 + 8 * q^27 - 18 * q^29 + 6 * q^35 - 6 * q^39 - 34 * q^43 - 8 * q^49 + 20 * q^51 - 10 * q^53 - 16 * q^55 - 20 * q^61 - 10 * q^65 + 6 * q^69 - 34 * q^75 + 4 * q^77 + 2 * q^79 + 8 * q^81 - 18 * q^87 + 6 * q^91 + 78 * q^95

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/4368\mathbb{Z}\right)^\times$.

$n$	$1093$	$1249$	$1457$	$2017$	$3823$
$\chi(n)$	$1$	$1$	$1$	$-1$	$1$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$			$a_n / n^{(k-1)/2}$			$ \alpha_n $			$ \theta_n $
$p$	$a_p$			$a_p / p^{(k-1)/2}$			$ \alpha_p$			$ \theta_p $

$2$		0			0
$2$		0			0
$3$	1.00000			0.577350
$3$	1.00000			0.577350
$4$		0			0
$5$		−	2.94550i		−	1.31727i	−0.752464	−	0.658634i	$-0.771135\pi$
$5$		−	2.94550i		−	1.31727i	0.752464	−	0.658634i	$-0.228865\pi$
$6$		0			0
$7$		−	1.00000i		−	0.377964i
$7$		−	1.00000i		−	0.377964i
$8$		0			0
$9$	1.00000			0.333333
$10$		0			0
$11$			3.62146i			1.09191i	0.837814	+	0.545956i	$0.183833\pi$
$11$			3.62146i			1.09191i	−0.837814	+	0.545956i	$0.816167\pi$
$12$		0			0
$13$	−3.38801	+	1.23346i	−0.939664	+	0.342099i
$13$	−3.38801	+	1.23346i	−0.939664	+	0.342099i
$14$		0			0
$15$		−	2.94550i		−	0.760525i
$16$		0			0
$17$	1.53309			0.371829			0.185914	−	0.982566i	$-0.440475\pi$
$17$	1.53309			0.371829			0.185914	+	0.982566i	$0.440475\pi$
$18$		0			0
$19$			4.10005i			0.940616i	0.882502	+	0.470308i	$0.155857\pi$
$19$			4.10005i			0.940616i	−0.882502	+	0.470308i	$0.844143\pi$
$20$		0			0
$21$		−	1.00000i		−	0.218218i
$22$		0			0
$23$	−7.03387			−1.46666			−0.733332	−	0.679871i	$-0.762036\pi$
$23$	−7.03387			−1.46666			−0.733332	+	0.679871i	$0.762036\pi$
$24$		0			0
$25$	−3.67596			−0.735193
$26$		0			0
$27$	1.00000			0.192450
$28$		0			0
$29$	−3.79095			−0.703961			−0.351981	−	0.936007i	$-0.614492\pi$
$29$	−3.79095			−0.703961			−0.351981	+	0.936007i	$0.614492\pi$
$30$		0			0
$31$			1.79095i			0.321664i	0.986982	+	0.160832i	$0.0514177\pi$
$31$			1.79095i			0.321664i	−0.986982	+	0.160832i	$0.948582\pi$
$32$		0			0
$33$			3.62146i			0.630416i
$34$		0			0
$35$	−2.94550			−0.497880
$36$		0			0
$37$			7.24292i			1.19073i	0.803456	+	0.595365i	$0.202992\pi$
$37$			7.24292i			1.19073i	−0.803456	+	0.595365i	$0.797008\pi$
$38$		0			0
$39$	−3.38801	+	1.23346i	−0.542515	+	0.197511i
$40$		0			0
$41$			9.15455i			1.42970i	0.699277	+	0.714850i	$0.253505\pi$
$41$			9.15455i			1.42970i	−0.699277	+	0.714850i	$0.746495\pi$
$42$		0			0
$43$	−5.79095			−0.883111			−0.441556	−	0.897234i	$-0.645573\pi$
$43$	−5.79095			−0.883111			−0.441556	+	0.897234i	$0.645573\pi$
$44$		0			0
$45$		−	2.94550i		−	0.439089i
$46$		0			0
$47$		−	7.25460i		−	1.05819i	−0.848562	−	0.529096i	$-0.822531\pi$
$47$		−	7.25460i		−	1.05819i	0.848562	−	0.529096i	$-0.177469\pi$
$48$		0			0
$49$	−1.00000			−0.142857
$50$		0			0
$51$	1.53309			0.214676
$52$		0			0
$53$	−13.3430			−1.83280			−0.916399	−	0.400266i	$-0.868918\pi$
$53$	−13.3430			−1.83280			−0.916399	+	0.400266i	$0.868918\pi$
$54$		0			0
$55$	10.6670			1.43834
$56$		0			0
$57$			4.10005i			0.543065i
$58$		0			0
$59$			4.84545i			0.630824i	0.948955	+	0.315412i	$0.102143\pi$
$59$			4.84545i			0.630824i	−0.948955	+	0.315412i	$0.897857\pi$
$60$		0			0
$61$	2.77601			0.355432			0.177716	−	0.984082i	$-0.443129\pi$
$61$	2.77601			0.355432			0.177716	+	0.984082i	$0.443129\pi$
$62$		0			0
$63$		−	1.00000i		−	0.125988i
$64$		0			0
$65$	3.63314	+	9.97937i	0.450636	+	1.23779i
$66$		0			0
$67$		−	14.8248i		−	1.81114i	−0.424197	−	0.905570i	$-0.639444\pi$
$67$		−	14.8248i		−	1.81114i	0.424197	−	0.905570i	$-0.360556\pi$
$68$		0			0
$69$	−7.03387			−0.846778
$70$		0			0
$71$		−	3.62146i		−	0.429788i	−0.976637	−	0.214894i	$-0.931059\pi$
$71$		−	3.62146i		−	0.429788i	0.976637	−	0.214894i	$-0.0689407\pi$
$72$		0			0
$73$			14.5670i			1.70493i	0.522781	+	0.852467i	$0.324894\pi$
$73$			14.5670i			1.70493i	−0.522781	+	0.852467i	$0.675106\pi$
$74$		0			0
$75$	−3.67596			−0.424464
$76$		0			0
$77$	3.62146			0.412704
$78$		0			0
$79$	−6.56696			−0.738841			−0.369420	−	0.929262i	$-0.620444\pi$
$79$	−6.56696			−0.738841			−0.369420	+	0.929262i	$0.620444\pi$
$80$		0			0
$81$	1.00000			0.111111
$82$		0			0
$83$			7.25460i			0.796296i	0.917321	+	0.398148i	$0.130347\pi$
$83$			7.25460i			0.796296i	−0.917321	+	0.398148i	$0.869653\pi$
$84$		0			0
$85$		−	4.51571i		−	0.489798i
$86$		0			0
$87$	−3.79095			−0.406432
$88$		0			0
$89$		−	0.636396i		−	0.0674578i	−0.999431	−	0.0337289i	$-0.989262\pi$
$89$		−	0.636396i		−	0.0674578i	0.999431	−	0.0337289i	$-0.0107383\pi$
$90$		0			0
$91$	1.23346	+	3.38801i	0.129301	+	0.355160i
$92$		0			0
$93$			1.79095i			0.185713i
$94$		0			0
$95$	12.0767			1.23904
$96$		0			0
$97$		−	11.3240i		−	1.14978i	−0.818230	−	0.574891i	$-0.805044\pi$
$97$		−	11.3240i		−	1.14978i	0.818230	−	0.574891i	$-0.194956\pi$
$98$		0			0
$99$			3.62146i			0.363971i
$100$		0			0
$101$	−6.04880			−0.601879			−0.300939	−	0.953643i	$-0.597300\pi$
$101$	−6.04880			−0.601879			−0.300939	+	0.953643i	$0.597300\pi$
$102$		0			0
$103$	−18.6670			−1.83932			−0.919658	−	0.392721i	$-0.871534\pi$
$103$	−18.6670			−1.83932			−0.919658	+	0.392721i	$0.871534\pi$
$104$		0			0
$105$	−2.94550			−0.287451
$106$		0			0
$107$	4.82482			0.466433			0.233216	−	0.972425i	$-0.425075\pi$
$107$	4.82482			0.466433			0.233216	+	0.972425i	$0.425075\pi$
$108$		0			0
$109$			4.95718i			0.474811i	0.971411	+	0.237406i	$0.0762971\pi$
$109$			4.95718i			0.474811i	−0.971411	+	0.237406i	$0.923703\pi$
$110$		0			0
$111$			7.24292i			0.687468i
$112$		0			0
$113$	3.79095			0.356622			0.178311	−	0.983974i	$-0.442937\pi$
$113$	3.79095			0.356622			0.178311	+	0.983974i	$0.442937\pi$
$114$		0			0
$115$			20.7183i			1.93199i
$116$		0			0
$117$	−3.38801	+	1.23346i	−0.313221	+	0.114033i
$118$		0			0
$119$		−	1.53309i		−	0.140538i
$120$		0			0
$121$	−2.11498			−0.192271
$122$		0			0
$123$			9.15455i			0.825438i
$124$		0			0
$125$		−	3.89995i		−	0.348822i
$126$		0			0
$127$	10.4903			0.930861			0.465430	−	0.885085i	$-0.345900\pi$
$127$	10.4903			0.930861			0.465430	+	0.885085i	$0.345900\pi$
$128$		0			0
$129$	−5.79095			−0.509864
$130$		0			0
$131$	−5.35193			−0.467600			−0.233800	−	0.972285i	$-0.575116\pi$
$131$	−5.35193			−0.467600			−0.233800	+	0.972285i	$0.575116\pi$
$132$		0			0
$133$	4.10005			0.355519
$134$		0			0
$135$		−	2.94550i		−	0.253508i
$136$		0			0
$137$		−	11.6936i		−	0.999054i	−0.866298	−	0.499527i	$-0.833507\pi$
$137$		−	11.6936i		−	0.999054i	0.866298	−	0.499527i	$-0.166493\pi$
$138$		0			0
$139$	14.6670			1.24404			0.622020	−	0.783002i	$-0.286312\pi$
$139$	14.6670			1.24404			0.622020	+	0.783002i	$0.286312\pi$
$140$		0			0
$141$		−	7.25460i		−	0.610948i
$142$		0			0
$143$	−4.46691	−	12.2695i	−0.373542	−	1.02603i
$144$		0			0
$145$			11.1662i			0.927305i
$146$		0			0
$147$	−1.00000			−0.0824786
$148$		0			0
$149$			12.6274i			1.03448i	0.855840	+	0.517240i	$0.173041\pi$
$149$			12.6274i			1.03448i	−0.855840	+	0.517240i	$0.826959\pi$
$150$		0			0
$151$			18.4858i			1.50436i	0.658959	+	0.752178i	$0.270997\pi$
$151$			18.4858i			1.50436i	−0.658959	+	0.752178i	$0.729003\pi$
$152$		0			0
$153$	1.53309			0.123943
$154$		0			0
$155$	5.27523			0.423717
$156$		0			0
$157$	−1.06618			−0.0850904			−0.0425452	−	0.999095i	$-0.513547\pi$
$157$	−1.06618			−0.0850904			−0.0425452	+	0.999095i	$0.513547\pi$
$158$		0			0
$159$	−13.3430			−1.05817
$160$		0			0
$161$			7.03387i			0.554347i
$162$		0			0
$163$			23.4430i			1.83620i	0.396350	+	0.918100i	$0.370277\pi$
$163$			23.4430i			1.83620i	−0.396350	+	0.918100i	$0.629723\pi$
$164$		0			0
$165$	10.6670			0.830426
$166$		0			0
$167$		−	7.67271i		−	0.593732i	−0.954919	−	0.296866i	$-0.904059\pi$
$167$		−	7.67271i		−	0.593732i	0.954919	−	0.296866i	$-0.0959415\pi$
$168$		0			0
$169$	9.95718	−	8.35791i	0.765937	−	0.642916i
$170$		0			0
$171$			4.10005i			0.313539i
$172$		0			0
$173$	−2.66701			−0.202769			−0.101385	−	0.994847i	$-0.532327\pi$
$173$	−2.66701			−0.202769			−0.101385	+	0.994847i	$0.532327\pi$
$174$		0			0
$175$			3.67596i			0.277877i
$176$		0			0
$177$			4.84545i			0.364206i
$178$		0			0
$179$	11.2340			0.839666			0.419833	−	0.907601i	$-0.362089\pi$
$179$	11.2340			0.839666			0.419833	+	0.907601i	$0.362089\pi$
$180$		0			0
$181$	9.58189			0.712217			0.356108	−	0.934445i	$-0.384103\pi$
$181$	9.58189			0.712217			0.356108	+	0.934445i	$0.384103\pi$
$182$		0			0
$183$	2.77601			0.205209
$184$		0			0
$185$	21.3340			1.56851
$186$		0			0
$187$			5.55203i			0.406004i
$188$		0			0
$189$		−	1.00000i		−	0.0727393i
$190$		0			0
$191$	7.89100			0.570973			0.285486	−	0.958383i	$-0.407845\pi$
$191$	7.89100			0.570973			0.285486	+	0.958383i	$0.407845\pi$
$192$		0			0
$193$		−	5.89100i		−	0.424043i	−0.977265	−	0.212022i	$-0.931995\pi$
$193$		−	5.89100i		−	0.424043i	0.977265	−	0.212022i	$-0.0680047\pi$
$194$		0			0
$195$	3.63314	+	9.97937i	0.260175	+	0.714637i
$196$		0			0
$197$			19.1546i			1.36471i	0.731023	+	0.682353i	$0.239043\pi$
$197$			19.1546i			1.36471i	−0.731023	+	0.682353i	$0.760957\pi$
$198$		0			0
$199$	−8.20010			−0.581290			−0.290645	−	0.956831i	$-0.593870\pi$
$199$	−8.20010			−0.581290			−0.290645	+	0.956831i	$0.593870\pi$
$200$		0			0
$201$		−	14.8248i		−	1.04566i
$202$		0			0
$203$			3.79095i			0.266072i
$204$		0			0
$205$	26.9647			1.88330
$206$		0			0
$207$	−7.03387			−0.488888
$208$		0			0
$209$	−14.8482			−1.02707
$210$		0			0
$211$	11.7009			0.805522			0.402761	−	0.915305i	$-0.368051\pi$
$211$	11.7009			0.805522			0.402761	+	0.915305i	$0.368051\pi$
$212$		0			0
$213$		−	3.62146i		−	0.248138i
$214$		0			0
$215$			17.0572i			1.16329i
$216$		0			0
$217$	1.79095			0.121577
$218$		0			0
$219$			14.5670i			0.984344i
$220$		0			0
$221$	−5.19412	+	1.89100i	−0.349394	+	0.127202i
$222$		0			0
$223$			5.45198i			0.365091i	0.983197	+	0.182546i	$0.0584337\pi$
$223$			5.45198i			0.365091i	−0.983197	+	0.182546i	$0.941566\pi$
$224$		0			0
$225$	−3.67596			−0.245064
$226$		0			0
$227$		−	27.1133i		−	1.79957i	−0.436331	−	0.899786i	$-0.643722\pi$
$227$		−	27.1133i		−	1.79957i	0.436331	−	0.899786i	$-0.356278\pi$
$228$		0			0
$229$			17.4918i			1.15589i	0.816075	+	0.577946i	$0.196146\pi$
$229$			17.4918i			1.15589i	−0.816075	+	0.577946i	$0.803854\pi$
$230$		0			0
$231$	3.62146			0.238275
$232$		0			0
$233$	13.3430			0.874127			0.437064	−	0.899431i	$-0.356018\pi$
$233$	13.3430			0.874127			0.437064	+	0.899431i	$0.356018\pi$
$234$		0			0
$235$	−21.3684			−1.39392
$236$		0			0
$237$	−6.56696			−0.426570
$238$		0			0
$239$		−	12.7554i		−	0.825077i	−0.910940	−	0.412539i	$-0.864642\pi$
$239$		−	12.7554i		−	0.825077i	0.910940	−	0.412539i	$-0.135358\pi$
$240$		0			0
$241$			9.94875i			0.640856i	0.947273	+	0.320428i	$0.103827\pi$
$241$			9.94875i			0.640856i	−0.947273	+	0.320428i	$0.896173\pi$
$242$		0			0
$243$	1.00000			0.0641500
$244$		0			0
$245$			2.94550i			0.188181i
$246$		0			0
$247$	−5.05723	−	13.8910i	−0.321784	−	0.883863i
$248$		0			0
$249$			7.25460i			0.459742i
$250$		0			0
$251$	−24.4002			−1.54013			−0.770064	−	0.637967i	$-0.779776\pi$
$251$	−24.4002			−1.54013			−0.770064	+	0.637967i	$0.779776\pi$
$252$		0			0
$253$		−	25.4729i		−	1.60147i
$254$		0			0
$255$		−	4.51571i		−	0.282785i
$256$		0			0
$257$	−28.2190			−1.76026			−0.880128	−	0.474737i	$-0.842543\pi$
$257$	−28.2190			−1.76026			−0.880128	+	0.474737i	$0.842543\pi$
$258$		0			0
$259$	7.24292			0.450053
$260$		0			0
$261$	−3.79095			−0.234654
$262$		0			0
$263$	11.4520			0.706159			0.353080	−	0.935593i	$-0.385134\pi$
$263$	11.4520			0.706159			0.353080	+	0.935593i	$0.385134\pi$
$264$		0			0
$265$			39.3017i			2.41428i
$266$		0			0
$267$		−	0.636396i		−	0.0389468i
$268$		0			0
$269$	19.8009			1.20728			0.603642	−	0.797255i	$-0.293716\pi$
$269$	19.8009			1.20728			0.603642	+	0.797255i	$0.293716\pi$
$270$		0			0
$271$			12.5391i			0.761694i	0.924638	+	0.380847i	$0.124368\pi$
$271$			12.5391i			0.761694i	−0.924638	+	0.380847i	$0.875632\pi$
$272$		0			0
$273$	1.23346	+	3.38801i	0.0746521	+	0.205051i
$274$		0			0
$275$		−	13.3124i		−	0.802765i
$276$		0			0
$277$	−2.43902			−0.146547			−0.0732733	−	0.997312i	$-0.523345\pi$
$277$	−2.43902			−0.146547			−0.0732733	+	0.997312i	$0.523345\pi$
$278$		0			0
$279$			1.79095i			0.107221i
$280$		0			0
$281$		−	25.9794i		−	1.54980i	−0.632084	−	0.774900i	$-0.717800\pi$
$281$		−	25.9794i		−	1.54980i	0.632084	−	0.774900i	$-0.282200\pi$
$282$		0			0
$283$	7.78199			0.462592			0.231296	−	0.972883i	$-0.425703\pi$
$283$	7.78199			0.462592			0.231296	+	0.972883i	$0.425703\pi$
$284$		0			0
$285$	12.0767			0.715362
$286$		0			0
$287$	9.15455			0.540376
$288$		0			0
$289$	−14.6496			−0.861743
$290$		0			0
$291$		−	11.3240i		−	0.663827i
$292$		0			0
$293$		−	7.14560i		−	0.417450i	−0.977974	−	0.208725i	$-0.933069\pi$
$293$		−	7.14560i		−	0.417450i	0.977974	−	0.208725i	$-0.0669314\pi$
$294$		0			0
$295$	14.2723			0.830963
$296$		0			0
$297$			3.62146i			0.210139i
$298$		0			0
$299$	23.8308	−	8.67596i	1.37817	−	0.501744i
$300$		0			0
$301$			5.79095i			0.333785i
$302$		0			0
$303$	−6.04880			−0.347495
$304$		0			0
$305$		−	8.17674i		−	0.468199i
$306$		0			0
$307$			10.4092i			0.594082i	0.954865	+	0.297041i	$0.0959998\pi$
$307$			10.4092i			0.594082i	−0.954865	+	0.297041i	$0.904000\pi$
$308$		0			0
$309$	−18.6670			−1.06193
$310$		0			0
$311$	−4.51571			−0.256063			−0.128031	−	0.991770i	$-0.540866\pi$
$311$	−4.51571			−0.256063			−0.128031	+	0.991770i	$0.540866\pi$
$312$		0			0
$313$	−21.6242			−1.22227			−0.611136	−	0.791526i	$-0.709287\pi$
$313$	−21.6242			−1.22227			−0.611136	+	0.791526i	$0.709287\pi$
$314$		0			0
$315$	−2.94550			−0.165960
$316$		0			0
$317$		−	16.8275i		−	0.945129i	−0.881296	−	0.472565i	$-0.843328\pi$
$317$		−	16.8275i		−	0.945129i	0.881296	−	0.472565i	$-0.156672\pi$
$318$		0			0
$319$		−	13.7288i		−	0.768664i
$320$		0			0
$321$	4.82482			0.269295
$322$		0			0
$323$			6.28575i			0.349748i
$324$		0			0
$325$	12.4542	−	4.53413i	0.690834	−	0.251509i
$326$		0			0
$327$			4.95718i			0.274133i
$328$		0			0
$329$	−7.25460			−0.399959
$330$		0			0
$331$			36.1355i			1.98619i	0.117331	+	0.993093i	$0.462566\pi$
$331$			36.1355i			1.98619i	−0.117331	+	0.993093i	$0.537434\pi$
$332$		0			0
$333$			7.24292i			0.396910i
$334$		0			0
$335$	−43.6665			−2.38575
$336$		0			0
$337$	−33.4595			−1.82266			−0.911328	−	0.411681i	$-0.864942\pi$
$337$	−33.4595			−1.82266			−0.911328	+	0.411681i	$0.864942\pi$
$338$		0			0
$339$	3.79095			0.205896
$340$		0			0
$341$	−6.48585			−0.351228
$342$		0			0
$343$			1.00000i			0.0539949i
$344$		0			0
$345$			20.7183i			1.11543i
$346$		0			0
$347$	8.72721			0.468501			0.234251	−	0.972176i	$-0.424736\pi$
$347$	8.72721			0.468501			0.234251	+	0.972176i	$0.424736\pi$
$348$		0			0
$349$		−	0.852706i		−	0.0456443i	−0.999740	−	0.0228222i	$-0.992735\pi$
$349$		−	0.852706i		−	0.0456443i	0.999740	−	0.0228222i	$-0.00726515\pi$
$350$		0			0
$351$	−3.38801	+	1.23346i	−0.180838	+	0.0658370i
$352$		0			0
$353$		−	7.46365i		−	0.397250i	−0.980076	−	0.198625i	$-0.936352\pi$
$353$		−	7.46365i		−	0.397250i	0.980076	−	0.198625i	$-0.0636476\pi$
$354$		0			0
$355$	−10.6670			−0.566146
$356$		0			0
$357$		−	1.53309i		−	0.0811397i
$358$		0			0
$359$			12.4019i			0.654547i	0.944930	+	0.327274i	$0.106130\pi$
$359$			12.4019i			0.654547i	−0.944930	+	0.327274i	$0.893870\pi$
$360$		0			0
$361$	2.18959			0.115241
$362$		0			0
$363$	−2.11498			−0.111008
$364$		0			0
$365$	42.9070			2.24585
$366$		0			0
$367$	9.31508			0.486243			0.243122	−	0.969996i	$-0.421829\pi$
$367$	9.31508			0.486243			0.243122	+	0.969996i	$0.421829\pi$
$368$		0			0
$369$			9.15455i			0.476567i
$370$		0			0
$371$			13.3430i			0.692733i
$372$		0			0
$373$	−3.73319			−0.193297			−0.0966486	−	0.995319i	$-0.530812\pi$
$373$	−3.73319			−0.193297			−0.0966486	+	0.995319i	$0.530812\pi$
$374$		0			0
$375$		−	3.89995i		−	0.201393i
$376$		0			0
$377$	12.8438	−	4.67596i	0.661487	−	0.240824i
$378$		0			0
$379$		−	12.0000i		−	0.616399i	−0.951322	−	0.308199i	$-0.900274\pi$
$379$		−	12.0000i		−	0.616399i	0.951322	−	0.308199i	$-0.0997264\pi$
$380$		0			0
$381$	10.4903			0.537433
$382$		0			0
$383$			8.80419i			0.449873i	0.974373	+	0.224936i	$0.0722175\pi$
$383$			8.80419i			0.449873i	−0.974373	+	0.224936i	$0.927783\pi$
$384$		0			0
$385$		−	10.6670i		−	0.543641i
$386$		0			0
$387$	−5.79095			−0.294370
$388$		0			0
$389$	−26.4978			−1.34349			−0.671746	−	0.740781i	$-0.734455\pi$
$389$	−26.4978			−1.34349			−0.671746	+	0.740781i	$0.734455\pi$
$390$		0			0
$391$	−10.7836			−0.545348
$392$		0			0
$393$	−5.35193			−0.269969
$394$		0			0
$395$			19.3430i			0.973251i
$396$		0			0
$397$			16.0399i			0.805017i	0.915416	+	0.402509i	$0.131862\pi$
$397$			16.0399i			0.805017i	−0.915416	+	0.402509i	$0.868138\pi$
$398$		0			0
$399$	4.10005			0.205259
$400$		0			0
$401$		−	12.2885i		−	0.613657i	−0.951765	−	0.306829i	$-0.900732\pi$
$401$		−	12.2885i		−	0.613657i	0.951765	−	0.306829i	$-0.0992678\pi$
$402$		0			0
$403$	−2.20905	−	6.06774i	−0.110041	−	0.302256i
$404$		0			0
$405$		−	2.94550i		−	0.146363i
$406$		0			0
$407$	−26.2300			−1.30017
$408$		0			0
$409$			24.8114i			1.22685i	0.789754	+	0.613423i	$0.210208\pi$
$409$			24.8114i			1.22685i	−0.789754	+	0.613423i	$0.789792\pi$
$410$		0			0
$411$		−	11.6936i		−	0.576804i
$412$		0			0
$413$	4.84545			0.238429
$414$		0			0
$415$	21.3684			1.04893
$416$		0			0
$417$	14.6670			0.718247
$418$		0			0
$419$	−21.6496			−1.05765			−0.528827	−	0.848730i	$-0.677368\pi$
$419$	−21.6496			−1.05765			−0.528827	+	0.848730i	$0.677368\pi$
$420$		0			0
$421$		−	22.4679i		−	1.09502i	−0.836799	−	0.547510i	$-0.815576\pi$
$421$		−	22.4679i		−	1.09502i	0.836799	−	0.547510i	$-0.184424\pi$
$422$		0			0
$423$		−	7.25460i		−	0.352731i
$424$		0			0
$425$	−5.63558			−0.273366
$426$		0			0
$427$		−	2.77601i		−	0.134341i
$428$		0			0
$429$	−4.46691	−	12.2695i	−0.215664	−	0.592379i
$430$		0			0
$431$		−	28.7787i		−	1.38622i	−0.720830	−	0.693112i	$-0.756239\pi$
$431$		−	28.7787i		−	1.38622i	0.720830	−	0.693112i	$-0.243761\pi$
$432$		0			0
$433$	−29.5217			−1.41872			−0.709361	−	0.704845i	$-0.751016\pi$
$433$	−29.5217			−1.41872			−0.709361	+	0.704845i	$0.751016\pi$
$434$		0			0
$435$			11.1662i			0.535380i
$436$		0			0
$437$		−	28.8392i		−	1.37957i
$438$		0			0
$439$	−18.6670			−0.890928			−0.445464	−	0.895300i	$-0.646961\pi$
$439$	−18.6670			−0.890928			−0.445464	+	0.895300i	$0.646961\pi$
$440$		0			0
$441$	−1.00000			−0.0476190
$442$		0			0
$443$	−15.2340			−0.723788			−0.361894	−	0.932219i	$-0.617870\pi$
$443$	−15.2340			−0.723788			−0.361894	+	0.932219i	$0.617870\pi$
$444$		0			0
$445$	−1.87450			−0.0888599
$446$		0			0
$447$			12.6274i			0.597258i
$448$		0			0
$449$			11.2755i			0.532125i	0.963956	+	0.266062i	$0.0857227\pi$
$449$			11.2755i			0.532125i	−0.963956	+	0.266062i	$0.914277\pi$
$450$		0			0
$451$	−33.1529			−1.56111
$452$		0			0
$453$			18.4858i			0.868541i
$454$		0			0
$455$	9.97937	−	3.63314i	0.467840	−	0.170324i
$456$		0			0
$457$		−	24.0234i		−	1.12377i	−0.827217	−	0.561883i	$-0.810077\pi$
$457$		−	24.0234i		−	1.12377i	0.827217	−	0.561883i	$-0.189923\pi$
$458$		0			0
$459$	1.53309			0.0715585
$460$		0			0
$461$			1.91163i			0.0890334i	0.999009	+	0.0445167i	$0.0141748\pi$
$461$			1.91163i			0.0890334i	−0.999009	+	0.0445167i	$0.985825\pi$
$462$		0			0
$463$		−	15.2250i		−	0.707567i	−0.935327	−	0.353783i	$-0.884895\pi$
$463$		−	15.2250i		−	0.707567i	0.935327	−	0.353783i	$-0.115105\pi$
$464$		0			0
$465$	5.27523			0.244633
$466$		0			0
$467$	−31.0483			−1.43674			−0.718371	−	0.695660i	$-0.755112\pi$
$467$	−31.0483			−1.43674			−0.718371	+	0.695660i	$0.755112\pi$
$468$		0			0
$469$	−14.8248			−0.684546
$470$		0			0
$471$	−1.06618			−0.0491270
$472$		0			0
$473$		−	20.9717i		−	0.964279i
$474$		0			0
$475$		−	15.0716i		−	0.691534i
$476$		0			0
$477$	−13.3430			−0.610933
$478$		0			0
$479$		−	39.6314i		−	1.81081i	−0.424552	−	0.905403i	$-0.639568\pi$
$479$		−	39.6314i		−	1.81081i	0.424552	−	0.905403i	$-0.360432\pi$
$480$		0			0
$481$	−8.93382	−	24.5391i	−0.407347	−	1.11889i
$482$		0			0
$483$			7.03387i			0.320052i
$484$		0			0
$485$	−33.3549			−1.51457
$486$		0			0
$487$		−	5.91435i		−	0.268005i	−0.990981	−	0.134002i	$-0.957217\pi$
$487$		−	5.91435i		−	0.268005i	0.990981	−	0.134002i	$-0.0427830\pi$
$488$		0			0
$489$			23.4430i			1.06013i
$490$		0			0
$491$	40.6068			1.83256			0.916280	−	0.400539i	$-0.131177\pi$
$491$	40.6068			1.83256			0.916280	+	0.400539i	$0.131177\pi$
$492$		0			0
$493$	−5.81186			−0.261753
$494$		0			0
$495$	10.6670			0.479446
$496$		0			0
$497$	−3.62146			−0.162445
$498$		0			0
$499$		−	27.9821i		−	1.25265i	−0.779562	−	0.626325i	$-0.784558\pi$
$499$		−	27.9821i		−	1.25265i	0.779562	−	0.626325i	$-0.215442\pi$
$500$		0			0
$501$		−	7.67271i		−	0.342791i
$502$		0			0
$503$	26.9458			1.20145			0.600727	−	0.799455i	$-0.294878\pi$
$503$	26.9458			1.20145			0.600727	+	0.799455i	$0.294878\pi$
$504$		0			0
$505$			17.8167i			0.792835i
$506$		0			0
$507$	9.95718	−	8.35791i	0.442214	−	0.371188i
$508$		0			0
$509$			0.836496i			0.0370770i	0.999828	+	0.0185385i	$0.00590133\pi$
$509$			0.836496i			0.0370770i	−0.999828	+	0.0185385i	$0.994099\pi$
$510$		0			0
$511$	14.5670			0.644404
$512$		0			0
$513$			4.10005i			0.181022i
$514$		0			0
$515$			54.9837i			2.42287i
$516$		0			0
$517$	26.2723			1.15545
$518$		0			0
$519$	−2.66701			−0.117069
$520$		0			0
$521$	22.3047			0.977186			0.488593	−	0.872512i	$-0.337510\pi$
$521$	22.3047			0.977186			0.488593	+	0.872512i	$0.337510\pi$
$522$		0			0
$523$	−28.0010			−1.22440			−0.612200	−	0.790703i	$-0.709715\pi$
$523$	−28.0010			−1.22440			−0.612200	+	0.790703i	$0.709715\pi$
$524$		0			0
$525$			3.67596i			0.160432i
$526$		0			0
$527$			2.74568i			0.119604i
$528$		0			0
$529$	26.4753			1.15110
$530$		0			0
$531$			4.84545i			0.210275i
$532$		0			0
$533$	−11.2917	−	31.0157i	−0.489099	−	1.34344i
$534$		0			0
$535$		−	14.2115i		−	0.614416i
$536$		0			0
$537$	11.2340			0.484782
$538$		0			0
$539$		−	3.62146i		−	0.155987i
$540$		0			0
$541$			5.86764i			0.252270i	0.992013	+	0.126135i	$0.0402572\pi$
$541$			5.86764i			0.252270i	−0.992013	+	0.126135i	$0.959743\pi$
$542$		0			0
$543$	9.58189			0.411198
$544$		0			0
$545$	14.6014			0.625454
$546$		0			0
$547$	−26.5068			−1.13335			−0.566674	−	0.823942i	$-0.691770\pi$
$547$	−26.5068			−1.13335			−0.566674	+	0.823942i	$0.691770\pi$
$548$		0			0
$549$	2.77601			0.118477
$550$		0			0
$551$		−	15.5431i		−	0.662157i
$552$		0			0
$553$			6.56696i			0.279256i
$554$		0			0
$555$	21.3340			0.905579
$556$		0			0
$557$			32.5862i			1.38072i	0.723466	+	0.690360i	$0.242548\pi$
$557$			32.5862i			1.38072i	−0.723466	+	0.690360i	$0.757452\pi$
$558$		0			0
$559$	19.6198	−	7.14287i	0.829828	−	0.302111i
$560$		0			0
$561$			5.55203i			0.234407i
$562$		0			0
$563$	9.71425			0.409407			0.204703	−	0.978824i	$-0.434377\pi$
$563$	9.71425			0.409407			0.204703	+	0.978824i	$0.434377\pi$
$564$		0			0
$565$		−	11.1662i		−	0.469767i
$566$		0			0
$567$		−	1.00000i		−	0.0419961i
$568$		0			0
$569$	3.79095			0.158925			0.0794624	−	0.996838i	$-0.474680\pi$
$569$	3.79095			0.158925			0.0794624	+	0.996838i	$0.474680\pi$
$570$		0			0
$571$	3.76550			0.157581			0.0787906	−	0.996891i	$-0.474894\pi$
$571$	3.76550			0.157581			0.0787906	+	0.996891i	$0.474894\pi$
$572$		0			0
$573$	7.89100			0.329651
$574$		0			0
$575$	25.8562			1.07828
$576$		0			0
$577$			10.7228i			0.446396i	0.974773	+	0.223198i	$0.0716496\pi$
$577$			10.7228i			0.446396i	−0.974773	+	0.223198i	$0.928350\pi$
$578$		0			0
$579$		−	5.89100i		−	0.244822i
$580$		0			0
$581$	7.25460			0.300972
$582$		0			0
$583$		−	48.3211i		−	2.00125i
$584$		0			0
$585$	3.63314	+	9.97937i	0.150212	+	0.412596i
$586$		0			0
$587$		−	34.0794i		−	1.40661i	−0.710889	−	0.703304i	$-0.751707\pi$
$587$		−	34.0794i		−	1.40661i	0.710889	−	0.703304i	$-0.248293\pi$
$588$		0			0
$589$	−7.34297			−0.302562
$590$		0			0
$591$			19.1546i			0.787913i
$592$		0			0
$593$		−	38.9600i		−	1.59990i	−0.600069	−	0.799948i	$-0.704860\pi$
$593$		−	38.9600i		−	1.59990i	0.600069	−	0.799948i	$-0.295140\pi$
$594$		0			0
$595$	−4.51571			−0.185126
$596$		0			0
$597$	−8.20010			−0.335608
$598$		0			0
$599$	4.90151			0.200270			0.100135	−	0.994974i	$-0.468072\pi$
$599$	4.90151			0.200270			0.100135	+	0.994974i	$0.468072\pi$
$600$		0			0
$601$	9.73970			0.397291			0.198645	−	0.980071i	$-0.436346\pi$
$601$	9.73970			0.397291			0.198645	+	0.980071i	$0.436346\pi$
$602$		0			0
$603$		−	14.8248i		−	0.603713i
$604$		0			0
$605$			6.22968i			0.253273i
$606$		0			0
$607$	17.8308			0.723730			0.361865	−	0.932231i	$-0.382140\pi$
$607$	17.8308			0.723730			0.361865	+	0.932231i	$0.382140\pi$
$608$		0			0
$609$			3.79095i			0.153617i
$610$		0			0
$611$	8.94823	+	24.5786i	0.362006	+	0.994345i
$612$		0			0
$613$			7.83777i			0.316565i	0.987394	+	0.158282i	$0.0505956\pi$
$613$			7.83777i			0.316565i	−0.987394	+	0.158282i	$0.949404\pi$
$614$		0			0
$615$	26.9647			1.08732
$616$		0			0
$617$			40.0471i			1.61224i	0.591755	+	0.806118i	$0.298435\pi$
$617$			40.0471i			1.61224i	−0.591755	+	0.806118i	$0.701565\pi$
$618$		0			0
$619$			9.07269i			0.364662i	0.983237	+	0.182331i	$0.0583643\pi$
$619$			9.07269i			0.364662i	−0.983237	+	0.182331i	$0.941636\pi$
$620$		0			0
$621$	−7.03387			−0.282259
$622$		0			0
$623$	−0.636396			−0.0254967
$624$		0			0
$625$	−29.8671			−1.19468
$626$		0			0
$627$	−14.8482			−0.592979
$628$		0			0
$629$			11.1041i			0.442748i
$630$		0			0
$631$		−	9.11056i		−	0.362686i	−0.983420	−	0.181343i	$-0.941956\pi$
$631$		−	9.11056i		−	0.362686i	0.983420	−	0.181343i	$-0.0580444\pi$
$632$		0			0
$633$	11.7009			0.465068
$634$		0			0
$635$		−	30.8991i		−	1.22619i
$636$		0			0
$637$	3.38801	−	1.23346i	0.134238	−	0.0488713i
$638$		0			0
$639$		−	3.62146i		−	0.143263i
$640$		0			0
$641$	22.8392			0.902095			0.451048	−	0.892500i	$-0.351050\pi$
$641$	22.8392			0.902095			0.451048	+	0.892500i	$0.351050\pi$
$642$		0			0
$643$			15.6844i			0.618532i	0.950976	+	0.309266i	$0.100083\pi$
$643$			15.6844i			0.618532i	−0.950976	+	0.309266i	$0.899917\pi$
$644$		0			0
$645$			17.0572i			0.671628i
$646$		0			0
$647$	47.6198			1.87213			0.936063	−	0.351832i	$-0.114441\pi$
$647$	47.6198			1.87213			0.936063	+	0.351832i	$0.114441\pi$
$648$		0			0
$649$	−17.5476			−0.688804
$650$		0			0
$651$	1.79095			0.0701928
$652$		0			0
$653$	42.4002			1.65925			0.829624	−	0.558322i	$-0.188555\pi$
$653$	42.4002			1.65925			0.829624	+	0.558322i	$0.188555\pi$
$654$		0			0
$655$			15.7641i			0.615954i
$656$		0			0
$657$			14.5670i			0.568311i
$658$		0			0
$659$	−15.7497			−0.613521			−0.306760	−	0.951787i	$-0.599245\pi$
$659$	−15.7497			−0.613521			−0.306760	+	0.951787i	$0.599245\pi$
$660$		0			0
$661$			17.3684i			0.675553i	0.941226	+	0.337777i	$0.109675\pi$
$661$			17.3684i			0.675553i	−0.941226	+	0.337777i	$0.890325\pi$
$662$		0			0
$663$	−5.19412	+	1.89100i	−0.201723	+	0.0734403i
$664$		0			0
$665$		−	12.0767i		−	0.468314i
$666$		0			0
$667$	26.6650			1.03247
$668$		0			0
$669$			5.45198i			0.210786i
$670$		0			0
$671$			10.0532i			0.388100i
$672$		0			0
$673$	−35.6754			−1.37519			−0.687593	−	0.726096i	$-0.741333\pi$
$673$	−35.6754			−1.37519			−0.687593	+	0.726096i	$0.741333\pi$
$674$		0			0
$675$	−3.67596			−0.141488
$676$		0			0
$677$	−30.7228			−1.18077			−0.590386	−	0.807121i	$-0.701025\pi$
$677$	−30.7228			−1.18077			−0.590386	+	0.807121i	$0.701025\pi$
$678$		0			0
$679$	−11.3240			−0.434577
$680$		0			0
$681$		−	27.1133i		−	1.03898i
$682$		0			0
$683$			12.2018i			0.466889i	0.972370	+	0.233444i	$0.0749997\pi$
$683$			12.2018i			0.466889i	−0.972370	+	0.233444i	$0.925000\pi$
$684$		0			0
$685$	−34.4436			−1.31602
$686$		0			0
$687$			17.4918i			0.667355i
$688$		0			0
$689$	45.2061	−	16.4580i	1.72221	−	0.626998i
$690$		0			0
$691$			19.7198i			0.750177i	0.926989	+	0.375089i	$0.122388\pi$
$691$			19.7198i			0.750177i	−0.926989	+	0.375089i	$0.877612\pi$
$692$		0			0
$693$	3.62146			0.137568
$694$		0			0
$695$		−	43.2017i		−	1.63873i
$696$		0			0
$697$			14.0348i			0.531604i
$698$		0			0
$699$	13.3430			0.504678
$700$		0			0
$701$	−38.4211			−1.45115			−0.725573	−	0.688145i	$-0.758425\pi$
$701$	−38.4211			−1.45115			−0.725573	+	0.688145i	$0.758425\pi$
$702$		0			0
$703$	−29.6963			−1.12002
$704$		0			0
$705$	−21.3684			−0.804781
$706$		0			0
$707$			6.04880i			0.227489i
$708$		0			0
$709$			29.1573i			1.09502i	0.836798	+	0.547512i	$0.184425\pi$
$709$			29.1573i			1.09502i	−0.836798	+	0.547512i	$0.815575\pi$
$710$		0			0
$711$	−6.56696			−0.246280
$712$		0			0
$713$		−	12.5973i		−	0.471772i
$714$		0			0
$715$	−36.1399	+	13.1573i	−1.35156	+	0.492054i
$716$		0			0
$717$		−	12.7554i		−	0.476358i
$718$		0			0
$719$	32.5157			1.21263			0.606316	−	0.795224i	$-0.292647\pi$
$719$	32.5157			1.21263			0.606316	+	0.795224i	$0.292647\pi$
$720$		0			0
$721$			18.6670i			0.695196i
$722$		0			0
$723$			9.94875i			0.369998i
$724$		0			0
$725$	13.9354			0.517547
$726$		0			0
$727$	−19.9821			−0.741095			−0.370547	−	0.928814i	$-0.620830\pi$
$727$	−19.9821			−0.741095			−0.370547	+	0.928814i	$0.620830\pi$
$728$		0			0
$729$	1.00000			0.0370370
$730$		0			0
$731$	−8.87804			−0.328366
$732$		0			0
$733$			26.1956i			0.967555i	0.875191	+	0.483778i	$0.160736\pi$
$733$			26.1956i			0.967555i	−0.875191	+	0.483778i	$0.839264\pi$
$734$		0			0
$735$			2.94550i			0.108646i
$736$		0			0
$737$	53.6875			1.97760
$738$		0			0
$739$			14.6247i			0.537979i	0.963143	+	0.268989i	$0.0866897\pi$
$739$			14.6247i			0.537979i	−0.963143	+	0.268989i	$0.913310\pi$
$740$		0			0
$741$	−5.05723	−	13.8910i	−0.185782	−	0.510299i
$742$		0			0
$743$			15.4268i			0.565955i	0.959127	+	0.282977i	$0.0913222\pi$
$743$			15.4268i			0.565955i	−0.959127	+	0.282977i	$0.908678\pi$
$744$		0			0
$745$	37.1941			1.36269
$746$		0			0
$747$			7.25460i			0.265432i
$748$		0			0
$749$		−	4.82482i		−	0.176295i
$750$		0			0
$751$	−2.62716			−0.0958664			−0.0479332	−	0.998851i	$-0.515263\pi$
$751$	−2.62716			−0.0958664			−0.0479332	+	0.998851i	$0.515263\pi$
$752$		0			0
$753$	−24.4002			−0.889193
$754$		0			0
$755$	54.4500			1.98164
$756$		0			0
$757$	35.9454			1.30646			0.653228	−	0.757161i	$-0.273414\pi$
$757$	35.9454			1.30646			0.653228	+	0.757161i	$0.273414\pi$
$758$		0			0
$759$		−	25.4729i		−	0.924607i
$760$		0			0
$761$			24.1028i			0.873725i	0.899528	+	0.436862i	$0.143910\pi$
$761$			24.1028i			0.873725i	−0.899528	+	0.436862i	$0.856090\pi$
$762$		0			0
$763$	4.95718			0.179462
$764$		0			0
$765$		−	4.51571i		−	0.163266i
$766$		0			0
$767$	−5.97664	−	16.4164i	−0.215804	−	0.592762i
$768$		0			0
$769$		−	51.8831i		−	1.87095i	−0.353391	−	0.935476i	$-0.614971\pi$
$769$		−	51.8831i		−	1.87095i	0.353391	−	0.935476i	$-0.385029\pi$
$770$		0			0
$771$	−28.2190			−1.01628
$772$		0			0
$773$		−	30.8833i		−	1.11080i	−0.831585	−	0.555398i	$-0.812566\pi$
$773$		−	30.8833i		−	1.11080i	0.831585	−	0.555398i	$-0.187434\pi$
$774$		0			0
$775$		−	6.58346i		−	0.236485i
$776$		0			0
$777$	7.24292			0.259838
$778$		0			0
$779$	−37.5341			−1.34480
$780$		0			0
$781$	13.1150			0.469291
$782$		0			0
$783$	−3.79095			−0.135477
$784$		0			0
$785$			3.14043i			0.112087i
$786$		0			0
$787$			53.9877i			1.92445i	0.272252	+	0.962226i	$0.412232\pi$
$787$			53.9877i			1.92445i	−0.272252	+	0.962226i	$0.587768\pi$
$788$		0			0
$789$	11.4520			0.407701
$790$		0			0
$791$		−	3.79095i		−	0.134791i
$792$		0			0
$793$	−9.40515	+	3.42409i	−0.333987	+	0.121593i
$794$		0			0
$795$			39.3017i			1.39389i
$796$		0			0
$797$	4.69688			0.166372			0.0831860	−	0.996534i	$-0.473490\pi$
$797$	4.69688			0.166372			0.0831860	+	0.996534i	$0.473490\pi$
$798$		0			0
$799$		−	11.1220i		−	0.393467i
$800$		0			0
$801$		−	0.636396i		−	0.0224859i
$802$		0			0
$803$	−52.7537			−1.86164
$804$		0			0
$805$	20.7183			0.730223
$806$		0			0
$807$	19.8009			0.697026
$808$		0			0
$809$	−22.4390			−0.788914			−0.394457	−	0.918914i	$-0.629067\pi$
$809$	−22.4390			−0.788914			−0.394457	+	0.918914i	$0.629067\pi$
$810$		0			0
$811$			10.4858i			0.368208i	0.982907	+	0.184104i	$0.0589383\pi$
$811$			10.4858i			0.368208i	−0.982907	+	0.184104i	$0.941062\pi$
$812$		0			0
$813$			12.5391i			0.439764i
$814$		0			0
$815$	69.0514			2.41876
$816$		0			0
$817$		−	23.7432i		−	0.830669i
$818$		0			0
$819$	1.23346	+	3.38801i	0.0431004	+	0.118387i
$820$		0			0
$821$			36.7808i			1.28366i	0.766847	+	0.641830i	$0.221824\pi$
$821$			36.7808i			1.28366i	−0.766847	+	0.641830i	$0.778176\pi$
$822$		0			0
$823$	28.9717			1.00989			0.504945	−	0.863152i	$-0.331513\pi$
$823$	28.9717			1.00989			0.504945	+	0.863152i	$0.331513\pi$
$824$		0			0
$825$		−	13.3124i		−	0.463477i
$826$		0			0
$827$		−	1.73046i		−	0.0601741i	−0.999547	−	0.0300871i	$-0.990422\pi$
$827$		−	1.73046i		−	0.0601741i	0.999547	−	0.0300871i	$-0.00957846\pi$
$828$		0			0
$829$	33.1972			1.15299			0.576494	−	0.817101i	$-0.304420\pi$
$829$	33.1972			1.15299			0.576494	+	0.817101i	$0.304420\pi$
$830$		0			0
$831$	−2.43902			−0.0846087
$832$		0			0
$833$	−1.53309			−0.0531184
$834$		0			0
$835$	−22.6000			−0.782104
$836$		0			0
$837$			1.79095i			0.0619042i
$838$		0			0
$839$			8.48312i			0.292870i	0.989220	+	0.146435i	$0.0467799\pi$
$839$			8.48312i			0.292870i	−0.989220	+	0.146435i	$0.953220\pi$
$840$		0			0
$841$	−14.6287			−0.504439
$842$		0			0
$843$		−	25.9794i		−	0.894777i
$844$		0			0
$845$	−24.6182	−	29.3289i	−0.846892	−	1.00894i
$846$		0			0
$847$			2.11498i			0.0726717i
$848$		0			0
$849$	7.78199			0.267077
$850$		0			0
$851$		−	50.9458i		−	1.74640i
$852$		0			0
$853$		−	14.9617i		−	0.512279i	−0.966640	−	0.256140i	$-0.917549\pi$
$853$		−	14.9617i		−	0.512279i	0.966640	−	0.256140i	$-0.0824507\pi$
$854$		0			0
$855$	12.0767			0.413014
$856$		0			0
$857$	−46.1334			−1.57589			−0.787943	−	0.615748i	$-0.788854\pi$
$857$	−46.1334			−1.57589			−0.787943	+	0.615748i	$0.788854\pi$
$858$		0			0
$859$	−15.3042			−0.522171			−0.261085	−	0.965316i	$-0.584080\pi$
$859$	−15.3042			−0.522171			−0.261085	+	0.965316i	$0.584080\pi$
$860$		0			0
$861$	9.15455			0.311986
$862$		0			0
$863$		−	19.8449i		−	0.675529i	−0.941231	−	0.337764i	$-0.890329\pi$
$863$		−	19.8449i		−	0.675529i	0.941231	−	0.337764i	$-0.109671\pi$
$864$		0			0
$865$			7.85568i			0.267101i
$866$		0			0
$867$	−14.6496			−0.497528
$868$		0			0
$869$		−	23.7820i		−	0.806749i
$870$		0			0
$871$	18.2857	+	50.2266i	0.619589	+	1.70186i
$872$		0			0
$873$		−	11.3240i		−	0.383261i
$874$		0			0
$875$	−3.89995			−0.131842
$876$		0			0
$877$		−	13.5699i		−	0.458224i	−0.973400	−	0.229112i	$-0.926418\pi$
$877$		−	13.5699i		−	0.458224i	0.973400	−	0.229112i	$-0.0735822\pi$
$878$		0			0
$879$		−	7.14560i		−	0.241015i
$880$		0			0
$881$	25.5331			0.860232			0.430116	−	0.902774i	$-0.358473\pi$
$881$	25.5331			0.860232			0.430116	+	0.902774i	$0.358473\pi$
$882$		0			0
$883$	0.102030			0.00343357			0.00171679	−	0.999999i	$-0.499454\pi$
$883$	0.102030			0.00343357			0.00171679	+	0.999999i	$0.499454\pi$
$884$		0			0
$885$	14.2723			0.479757
$886$		0			0
$887$	−25.1339			−0.843914			−0.421957	−	0.906616i	$-0.638657\pi$
$887$	−25.1339			−0.843914			−0.421957	+	0.906616i	$0.638657\pi$
$888$		0			0
$889$		−	10.4903i		−	0.351832i
$890$		0			0
$891$			3.62146i			0.121324i
$892$		0			0
$893$	29.7442			0.995353
$894$		0			0
$895$		−	33.0896i		−	1.10606i
$896$		0			0
$897$	23.8308	−	8.67596i	0.795687	−	0.289682i
$898$		0			0
$899$		−	6.78939i		−	0.226439i
$900$		0			0
$901$	−20.4560			−0.681487
$902$		0			0
$903$			5.79095i			0.192711i
$904$		0			0
$905$		−	28.2235i		−	0.938179i
$906$		0			0
$907$	20.5789			0.683312			0.341656	−	0.939825i	$-0.389012\pi$
$907$	20.5789			0.683312			0.341656	+	0.939825i	$0.389012\pi$
$908$		0			0
$909$	−6.04880			−0.200626
$910$		0			0
$911$	−23.1782			−0.767928			−0.383964	−	0.923348i	$-0.625441\pi$
$911$	−23.1782			−0.767928			−0.383964	+	0.923348i	$0.625441\pi$
$912$		0			0
$913$	−26.2723			−0.869485
$914$		0			0
$915$		−	8.17674i		−	0.270315i
$916$		0			0
$917$			5.35193i			0.176736i
$918$		0			0
$919$	−10.7083			−0.353233			−0.176617	−	0.984280i	$-0.556515\pi$
$919$	−10.7083			−0.353233			−0.176617	+	0.984280i	$0.556515\pi$
$920$		0			0
$921$			10.4092i			0.342993i
$922$		0			0
$923$	4.46691	+	12.2695i	0.147030	+	0.403857i
$924$		0			0
$925$		−	26.6247i		−	0.875415i
$926$		0			0
$927$	−18.6670			−0.613105
$928$		0			0
$929$		−	46.0083i		−	1.50948i	−0.656022	−	0.754742i	$-0.727762\pi$
$929$		−	46.0083i		−	1.50948i	0.656022	−	0.754742i	$-0.272238\pi$
$930$		0			0
$931$		−	4.10005i		−	0.134374i
$932$		0			0
$933$	−4.51571			−0.147838
$934$		0			0
$935$	16.3535			0.534816
$936$		0			0
$937$	−24.4256			−0.797951			−0.398976	−	0.916962i	$-0.630634\pi$
$937$	−24.4256			−0.797951			−0.398976	+	0.916962i	$0.630634\pi$
$938$		0			0
$939$	−21.6242			−0.705679
$940$		0			0
$941$			41.5757i			1.35533i	0.735372	+	0.677664i	$0.237008\pi$
$941$			41.5757i			1.35533i	−0.735372	+	0.677664i	$0.762992\pi$
$942$		0			0
$943$		−	64.3919i		−	2.09689i
$944$		0			0
$945$	−2.94550			−0.0958171
$946$		0			0
$947$		−	14.6697i		−	0.476702i	−0.971179	−	0.238351i	$-0.923393\pi$
$947$		−	14.6697i		−	0.476702i	0.971179	−	0.238351i	$-0.0766069\pi$
$948$		0			0
$949$	−17.9677	−	49.3530i	−0.583256	−	1.60206i
$950$		0			0
$951$		−	16.8275i		−	0.545671i
$952$		0			0
$953$	47.5083			1.53895			0.769473	−	0.638680i	$-0.220519\pi$
$953$	47.5083			1.53895			0.769473	+	0.638680i	$0.220519\pi$
$954$		0			0
$955$		−	23.2429i		−	0.752123i
$956$		0			0
$957$		−	13.7288i		−	0.443788i
$958$		0			0
$959$	−11.6936			−0.377607
$960$		0			0
$961$	27.7925			0.896533
$962$		0			0
$963$	4.82482			0.155478
$964$		0			0
$965$	−17.3519			−0.558578
$966$		0			0
$967$		−	11.0249i		−	0.354537i	−0.984162	−	0.177269i	$-0.943274\pi$
$967$		−	11.0249i		−	0.354537i	0.984162	−	0.177269i	$-0.0567262\pi$
$968$		0			0
$969$			6.28575i			0.201927i
$970$		0			0
$971$	−34.4858			−1.10670			−0.553352	−	0.832948i	$-0.686651\pi$
$971$	−34.4858			−1.10670			−0.553352	+	0.832948i	$0.686651\pi$
$972$		0			0
$973$		−	14.6670i		−	0.470203i
$974$		0			0
$975$	12.4542	−	4.53413i	0.398853	−	0.145209i
$976$		0			0
$977$		−	46.5797i		−	1.49022i	−0.666944	−	0.745108i	$-0.732398\pi$
$977$		−	46.5797i		−	1.49022i	0.666944	−	0.745108i	$-0.267602\pi$
$978$		0			0
$979$	2.30468			0.0736580
$980$		0			0
$981$			4.95718i			0.158270i
$982$		0			0
$983$			47.9704i			1.53002i	0.644019	+	0.765009i	$0.277266\pi$
$983$			47.9704i			1.53002i	−0.644019	+	0.765009i	$0.722734\pi$
$984$		0			0
$985$	56.4197			1.79768
$986$		0			0
$987$	−7.25460			−0.230917
$988$		0			0
$989$	40.7328			1.29523
$990$		0			0
$991$	53.1160			1.68729			0.843643	−	0.536905i	$-0.180407\pi$
$991$	53.1160			1.68729			0.843643	+	0.536905i	$0.180407\pi$
$992$		0			0
$993$			36.1355i			1.14672i
$994$		0			0
$995$			24.1534i			0.765714i
$996$		0			0
$997$	−56.7238			−1.79646			−0.898231	−	0.439524i	$-0.855147\pi$
$997$	−56.7238			−1.79646			−0.898231	+	0.439524i	$0.855147\pi$
$998$		0			0
$999$			7.24292i			0.229156i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4368.2.h.q.337.3		8
4.3	odd	2		273.2.c.c.64.4	✓	8
12.11	even	2		819.2.c.d.64.5		8
13.12	even	2	inner	4368.2.h.q.337.6		8
28.27	even	2		1911.2.c.l.883.4		8
52.31	even	4		3549.2.a.x.1.2		4
52.47	even	4		3549.2.a.v.1.3		4
52.51	odd	2		273.2.c.c.64.5	yes	8
156.155	even	2		819.2.c.d.64.4		8
364.363	even	2		1911.2.c.l.883.5		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
273.2.c.c.64.4	✓	8	4.3	odd	2
273.2.c.c.64.5	yes	8	52.51	odd	2
819.2.c.d.64.4		8	156.155	even	2
819.2.c.d.64.5		8	12.11	even	2
1911.2.c.l.883.4		8	28.27	even	2
1911.2.c.l.883.5		8	364.363	even	2
3549.2.a.v.1.3		4	52.47	even	4
3549.2.a.x.1.2		4	52.31	even	4
4368.2.h.q.337.3		8	1.1	even	1	trivial
4368.2.h.q.337.6		8	13.12	even	2	inner

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 4368 = 2^{4} \cdot 3 \cdot 7 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4368.h (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+1.00000 q^{3} -2.94550i q^{5} -1.00000i q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{3} -2.94550i q^{5} -1.00000i q^{7} +1.00000 q^{9} +3.62146i q^{11} +(-3.38801 + 1.23346i) q^{13} -2.94550i q^{15} +1.53309 q^{17} +4.10005i q^{19} -1.00000i q^{21} -7.03387 q^{23} -3.67596 q^{25} +1.00000 q^{27} -3.79095 q^{29} +1.79095i q^{31} +3.62146i q^{33} -2.94550 q^{35} +7.24292i q^{37} +(-3.38801 + 1.23346i) q^{39} +9.15455i q^{41} -5.79095 q^{43} -2.94550i q^{45} -7.25460i q^{47} -1.00000 q^{49} +1.53309 q^{51} -13.3430 q^{53} +10.6670 q^{55} +4.10005i q^{57} +4.84545i q^{59} +2.77601 q^{61} -1.00000i q^{63} +(3.63314 + 9.97937i) q^{65} -14.8248i q^{67} -7.03387 q^{69} -3.62146i q^{71} +14.5670i q^{73} -3.67596 q^{75} +3.62146 q^{77} -6.56696 q^{79} +1.00000 q^{81} +7.25460i q^{83} -4.51571i q^{85} -3.79095 q^{87} -0.636396i q^{89} +(1.23346 + 3.38801i) q^{91} +1.79095i q^{93} +12.0767 q^{95} -11.3240i q^{97} +3.62146i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 8 q^{3} + 8 q^{9}+O(q^{10}) \) 8 * q + 8 * q^3 + 8 * q^9	\( 8 q + 8 q^{3} + 8 q^{9} - 6 q^{13} + 20 q^{17} + 6 q^{23} - 34 q^{25} + 8 q^{27} - 18 q^{29} + 6 q^{35} - 6 q^{39} - 34 q^{43} - 8 q^{49} + 20 q^{51} - 10 q^{53} - 16 q^{55} - 20 q^{61} - 10 q^{65} + 6 q^{69} - 34 q^{75} + 4 q^{77} + 2 q^{79} + 8 q^{81} - 18 q^{87} + 6 q^{91} + 78 q^{95}+O(q^{100}) \) 8 * q + 8 * q^3 + 8 * q^9 - 6 * q^13 + 20 * q^17 + 6 * q^23 - 34 * q^25 + 8 * q^27 - 18 * q^29 + 6 * q^35 - 6 * q^39 - 34 * q^43 - 8 * q^49 + 20 * q^51 - 10 * q^53 - 16 * q^55 - 20 * q^61 - 10 * q^65 + 6 * q^69 - 34 * q^75 + 4 * q^77 + 2 * q^79 + 8 * q^81 - 18 * q^87 + 6 * q^91 + 78 * q^95

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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